Capital Asset Pricing Model

1 Introduction

In this handout we develop a model that can be used to determine how an investor can choosean optimal asset portfolio in this sense:

• the investor will earn the highest possible expected return given the level of volatility theinvestor is willing to accept or, equivalently,

• the investor’s portfolio will have the lowest level of volatility given the level of expectedreturn the investor requires.

The techniques used are called mean-variance optimization and the underlying theory is calledthe Capital Asset Pricing Model (CAPM). Under the assumptions of CAPM, it is possible todetermine the expected “risk-adjusted” return of any asset/security, which incorporates thesecurity’s expected return, volatility and its correlation with the “market portfolio.”

2 Mean-Variance Efficient Frontier

2.1 Setup

We consider a market with n risky assets, i = 1, 2, . . . , n and a risk-free asset labeled 0. Aninvestor wishes to invest B dollars in this market. Let

• Bi, i = 0, 1, 2, . . . , n, denote the allocation of the budget to asset i so that∑ni=0Bi = B,

• xi := Bi/B denote the portfolio weight of asset i, namely, the fraction of the investor’sbudget allocated to asset i,

• Ri denote the random one-period return on asset i, i = 1, 2, . . . , n, and let

• rf denote the risk-free return.

Example 1 Consider a portfolio of 200 shares of firm A worth $30/share and 100 shares offirm B worth $40/share. The total value of the portfolio is

200 ∗ ($30) + 100($40) = $10, 000. (1)

The respective portfolio weights are

xA =200 ∗ ($30)

$10, 000= 60%, xB =

100 ∗ ($40)$10, 000

= 40%. (2)

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2.2 Portfolio return

The investor’s one-period return on his/her portfolio is given by

One-period return =∑ni=0BiRiB

=n∑i=0

BiB

Ri =n∑i=0

xiRi. (3)

We shall think of B as fixed and hereafter identify a portfolio of the n+ 1 assets with a vector

x = (x0, x1, x2, . . . , xn) such thatn∑i=0

xi = 1. (4)

The portfolio’s random return will be denoted by

RP = R(x) :=n∑i=1

xiRi. (5)

We shall use the symbols ‘x’ or P to refer to a portfolio.

Remark 1 When xi < 0 the holder of the portfolio is short-selling asset i. In these noteswe permit unlimited short-selling. In practice, however, there a limits to the magnitude ofshort-selling. If short-selling is not permitted, i.e., each xi is constrained to be non-negative,then the solution approach outlined in these notes does not directly apply, though the problemcan be easily solved with commercial software.

Example 2 You bought the portfolio of Example 1. Suppose firm A’s share price goes up to$36 and firm B’s share price falls to $38. What is the new value of the portfolio? What returndid it earn? After the price change, what are the new portfolio weights? The new value of theportfolio is

200 ∗ ($36) + 100($38) = $11, 000 (6)

for a gain of $1,000 or 10% return on investment. A’s return was 36/30 - 1 = 20% and B’sreturn was 38/40 - 1 = -5%. Since the initial portfolio weights were xA = 60% and xB = 40%,we can also compute the portfolio’s return as

RP = xARA + xBRB = 0.60(20%) + 0.40(−5%) = 10%. (7)

The new portfolio weights are

xA =200 ∗ ($36)

$11, 000= 65.45%, xB =

100 ∗ ($38)$11, 000

= 34.55%. (8)

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2.3 Expected portfolio return

The expected return of a portfolio P is given by

µ(x) := E[RP ] = E[∑i

xiRi] =∑i

E[xiRi] =∑i

xiE[Ri]. (9)

Example 3 You invest $1000 in stock A, $3000 in stock B. You expect a return of 10% forstock A and 18% in stock B. What is your portfolio’s expected return? Your portfolio weightsare xA = 1, 000/4, 000 = 25% and xB = 3, 000/4, 000 = 75%. Therefore,

E[RP ] = 0.25(10%) + 0.75(18%) = 16%. (10)

2.4 Portfolio optimization via expected utility

The investor’s end-of-period wealth is B(1 +R(x)). We assume the investor is risk-averse andthat his/her utility function U(·) is sufficiently differentiable and strictly concave. The investor’soptimal portfolio x∗ should maximize his/her expected utility of end-of-period wealth, as follows:

Investor’s problem: maxx:∑n

i=0xi=1

E[U(1 +R(x))], (11)

where, for convenience, we have normalized the budget to one.

The investor’s problem is a non-trivial one. One difficulty is to obtain the distribution ofR(x), which depends on x. In lieu of solving the investor’s problem directly, we shall approxi-mate the expected utility via a second-order Taylor series expansion about mean end-of-periodwealth given by 1 + E[R(x)] := 1 + µ(x).

For each realization of the random variable R(x), and ignoring higher-order terms,

U(1 +R(x)) ≈ U(1 + µ(x)) + [(1 +R(x))− (1 + µ(x))]U ′(1 + µ(x)) +1

2[(1 +R(x))− (1 + µ(x))]2U ′′(1 + µ(x)). (12)

Taking expectations of both sides of (12),

E[U(1 +R(x))] ≈ E[U(1 + µ(x))] + U ′(1 + µ(x))E[R(x)− µ(x)] +1

2U ′′(1 + µ(x))E[R(x)− µ(x)]2. (13)

Using the fact that

E[R(x)− µ(x)] = E[R(x)]− µ(x) = 0, (14)E[R(x)− µ(x)]2 = V ar[R(x)], (15)

we haveE[U(1 +R(x))] ≈ U(1 + µ(x)) +

12U ′′(1 + µ(x))V ar[R(x)]. (16)

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Assuming that the approximation of expected utility given in (16) is reasonably accurate,maximizing the left-hand side of (16) is (essentially) equivalent to maximizing theright-hand side of (16).

Now fix the value for mean return µ(x) say to R̄0; that is, consider only those portfolios xfor which µ(x) = R̄0. In words, the investor is fixing his/her desired expected portfolio return.Since the sign of U ′′(·) is negative, and since U(1 + µ(x)) and U ′′(1 + µ(x)) are constants, itfollows from the right-hand side of (16) that the investor’s optimal portfolio must

• must minimize the variance of the portfolio’s return subject to achieving the requiredexpected return or, equivalently,

• maximize the portfolio’s expected return subject to fixing the portfolio’s variance to anacceptable level.

Such a portfolio is called mean-variance efficient. The set of all such mean-variance efficientportfolios is called the mean-variance efficient frontier.

3 Portfolio Variance

In this section we only consider the risky assets i = 1, 2, . . . , n. Keep in mind each Riis a random variable.

3.1 Basic definitions and properties

We recall some basic definitions and properties from statistics. Let

R̄i := E[Ri] (17)V ar[Ri] := E(Ri − R̄i)2 = E[R2

i ]− (R̄i)2 (18)

σi :=√V ar[Ri] (19)

denote, respectively, the mean, variance and standard deviation of Ri.

Recall that the covariance between two random variables X and Y is defined as

Cov(X,Y ) = σXY := E[(X − E[X])(Y − E[Y ])] = E[XY ]− E[X]E[Y ], (20)

and the correlation between two random variables X and Y is defined as

Corr(X,Y ) = ρXY :=Cov(X,Y )σXσY

(21)

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Recall that a correlation value must always be a number between -1 and 1. Note that covariancecan be determined from correlation via

σXY = Cov(X,Y ) = Corr(X,Y )σXσY = ρXY σXσY . (22)

It follows directly from (20) that

• variance may be expressed in terms of covariance:

V ar(X) = Cov(X,X), (23)

• covariance is symmetric:

Cov(X,Y ) = Cov(Y,X), (24)

• covariance is bilinear :

Cov(n∑i=1

Xi, Y ) =n∑i=1

Cov(Xi, Y ), (25)

Cov(αX, Y ) = Cov(X,αY ) = αCov(X,Y ), α a real number. (26)

3.2 Variance formula

In what follows, we let

σij := Cov(Ri, Rj) = σiσjCorr(Ri, Rj) = σiσjρij (27)

and let Σ denote the n by n matrix whose (i, j)th entry is given by σij . The matrix Σ is calledthe covariance matrix. Due to (24), σij = σji, which implies that the covariance matrix issymmetric, namely, Σ = ΣT (here, the symbol ‘T’ denotes transpose).

Using (23) and repeated application of (25) and (26), the portfolio’s (return) variance is

V ar[R(x)] = V ar[n∑i=1

xiRi] (28)

= Cov(n∑i=1

xiRi,n∑j=1

xjRj) (29)

=∑i

xiCov(Ri,∑j

xjRj) (30)

=∑i

xi

∑j

xjCov(Ri, Rj)

(31)

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= xTΣx (32)

=n∑i=1

n∑j=1

σijxixj . (33)

(34)

3.3 Volatility of a two-asset portfolio

When n = 2 the portfolio’s variance is2∑i=1

2∑j=1

σijxixj = σ21x

21 + σ12x1x2 + σ21x2x1 + σ2

2x22 (35)

= σ21x

21 + σ2

2x22 + 2σ12x1x2 (36)

= σ21x

21 + σ2

2x22 + 2σ1σ2ρ12x1x2 (37)

Here, we have used the fact that the covariances between the two assets, namely, σ12 and σ21,are equal.

Remark 2 In matrix notation, the portfolio’s variance may be represented as

V ar[R(x)] = xTΣx (38)

=(x1 x2

)( σ21 σ12

σ21 σ22

)(x1

x2

)(39)

=(x1 x2

)( σ21x1 + σ12x2

σ21x1 + σ22x2

)(40)

= (x1σ21x1 + x1σ12x2) + (x2σ21x1 + s2σ

22x2) (41)

= σ21x

21 + σ2

2x22 + 2σ12x1x2. (42)

Example 4 Stock returns will be more highly correlated when they are similarly affected by thesame economic events. This is why stocks in the same industry tend to have highly correlatedreturns than stocks in somewhat different industries. Table 1 (Table 11.3, p. 336 in CorporateFinance by Berk and DeMarzo) provides some examples.

What is the covariance between the returns for Microsoft and Dell?

Cov(RM , RD) = σmσDρMD = (0.37)(0.50)(0.62) = 0.1147. (43)

What is the standard deviation of a portfolio with equal amounts invested in these two stocks?

V ar(RP ) = V ar(0.5Rm + 0.5RD) (44)= σ2

Mx2M + σ2

Dx2D + 2σMDxMxD (45)

= (0.37)2(0.5)2 + (0.5)2(0.5)2 + 2(0.1147) = 0.1541 (46)σP =

√0.1541 = 39.26%. (47)

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Table 1: Historical Annual Volatilities and Correlations for Selected Stocks (basedon monthly returns, 1996-2008).

Alaskan Southwest Ford General GeneralMicrosoft Dell Air Airlines Motor Motors Mills

Volatility (StDev) 37% 50% 38% 31% 42% 41% 18%Correlation with:

Microsoft 1.00 0.62 0.25 0.23 0.26 0.23 0.10Dell 0.62 1.00 0.19 0.21 0.31 0.28 0.07

Alaska Air 0.25 0.19 1.00 0.30 0.16 0.13 0.11Southwest Airlines 0.23 0.21 0.30 1.00 0.25 0.22 0.20

Ford Motor 0.26 0.31 0.16 0.25 1.00 0.62 0.07General Motors 0.23 0.28 0.13 0.22 0.62 1.00 0.02

General Mills 0.10 0.07 0.11 0.20 0.07 0.02 1.00

Example 5 Consider a portfolio of Intel and Coca-Cola stocks. Data shown in Table 2. Let x

Table 2:

Stock Expected Return VolatilityIntel 26% 50%

Coca-Cola 6% 25%

denote the weight on Intel and 1− x denote the weight on Coca-Cola. As x varies from 0 to 1,the portfolio’s mean return E[R(x)] and standard deviation StDev(R(x)) =

√V ar(R(x)) will

vary. We can plot the points {(StDev(R(x)), E[R(x)]} in the “(x,y) space”. This collectionof points traces out a curve. The shape of the curve critically depends on the correlation ρbetween these two stocks. In fact, we can trace out a collection of such curves, one for eachpossible value of ρ ∈ [−1, 1].

What is the shape of the curve when ρ = +1? ρ = −1? How about values for ρ in between?The expected return

E[R(x)] = 0.26x+ 0.06(1− x) = 0.06 + 0.20x (48)

is linear in x and does not depend on ρ. As for the variance:

General ρ : V ar(R(x)) = (0.5)2x2 + (0.25)2(1− x)2 + 2x(1− x)(0.50)(0.25)ρ (49)ρ = +1 : V ar(R(x)) = (0.5)2x2 + (0.25)2(1− x)2 + 2x(1− x)(0.50)(0.25) (50)

= (0.5x+ 0.25(1− x))2 (51)

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= (0.25 + 0.25x)2 (52)StDev(R(x)) = 0.25 + 0.25x, x ∈ [0, 1] (53)

ρ = −1 : V ar(R(x)) = (0.5)2x2 + (0.25)2(1− x)2 − 2x(1− x)(0.50)(0.25) (54)= (0.5x− 0.25(1− x))2 (55)= (0.75x− 0.25)2 (56)

StDev(R(x)) = |0.75x− 0.25|, x ∈ [0, 1]. (57)

Note how the variance is a quadratic function of x. Note further than when ρ = +1, the StDevis linear in x, whereas when ρ = −1, the StDev is “piecewise linear” and will actually be zerowhen x = 1/3!

Example 6 Consider the data of Example 5. Assume ρ = 0. Suppose you have $20,000 in cashto invest. You decide to short sell $10,000 worth of Coco-Cola stock and invest the proceedsfrom your short sale plus your $20,000 in Intel. What is the expected return and volatility ofyour portfolio?

Your short sale is a negative investment of $10,000 in Coca-Cola stock. You have invested$30,000 in Intel. Your total investment is still $20,000. Your portfolio weights are 1.5 or 150%in Intel and -0.5 or -50% in Coca-Cola. Thus, the expected portfolio return is

1.5(0.26) +−0.5(0.06) = 36%. (58)

and its variance is0.502(1.5)2 + 0.252(−0.5)2 = 0.578125 (59)

for a StDev of 76%. By short-selling you have dramatically increased the expected return butyou have also significantly increased the volatility.

3.4 Minimum variance portfolio of two stocks

By definition, the minimum variance portfolio achieves the lowest variance, regardless of ex-pected return. For two stocks, such as examined in Example 5, it is easy to solve for theminimum variance portfolio. (We shall see that it is also easy to solve for the minimum vari-ance portfolio for the general case.) From the two-stock portfolio variance formula (36) withx2 = 1− x1, the variance is

V ar(x1R1 + (1− x1)R2) = σ21x

21 + σ2

2(1− x1)2 + 2x1(1− x1)σ12. (60)

Since there are no constraints on the values for x, we can take the derivative with respect to xand set equal to zero:

0 :=d

dx1V ar(R(x)) = 2σ2

1x1 − 2(1− x1)σ22 − 2σ12(1− 2x1) (61)

= 2(σ21 + σ2

2 − 2σ12)x1 − (σ22 − σ12). (62)

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Thus,

x∗1 =σ2

2 − σ12

σ21 + σ2

2 − 2σ12(63)

x∗2 =σ2

1 − σ12

σ21 + σ2

2 − 2σ12. (64)

Remark 3 When the stock returns are uncorrelated, the covariance σ12 equals zero. Then,the optimal portfolio is

(x∗1, x∗2) =

(σ2

2

σ21 + σ2

2

,σ2

1

σ21 + σ2

2

). (65)

Example 7 Let’s return to the data of Example 5. Assume the correlation coefficient is 0.20.Then the covariance σ12 = (0.50)(0.25)(0.2) = 0.025. To minimize the variance, the optimalweight for the Intel stock is

(0.25)2 − 0.025(0.50)2 + (0.25)2 − 2(0.025)

=0.03750.2625

= 1/7, (66)

and

V ar(R(x∗)) = (0.50)2(1/7)2 + (0.25)2(6/7)2 + 2(1/7)(6/7)(0.025) = 0.05714 (67)StDev(R(x∗)) =

√0.05714 = 23.9%. (68)

Note how the minimum volatility is less than the volatility of either stock!

3.5 Volatility of a large portfolio: the benefits of diversification

Let AvgV ar denote the average variance of the individual stocks, and let AvgCov denote theaverage covariance between the stocks. Imagine a world where the variance of each stock wasconstant and equal to AvgV ar and the covariance between stocks was constant and equal toAvgCov. Now consider an equally-weighted portfolio of theses stocks in which xi = 1/n foreach i = 1, 2, . . . , n. What is the variance of this equally-weighted portfolio? The covariancematrix Σ has n2 entries, n along the diagonal and n2 − n = n(n− 1) off-diagonal entries. Theentries along the diagonal correspond to each stock’s variance which equals AvgV ar, whereasthe off-diagonal entries correspond to the covariances which each equal AvgCov. Therefore, inthis special setting

V ar(RP ) =n∑i=1

n∑j=1

σijxixj (69)

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= (1/n2) ∗n∑i=1

n∑j=1

σij (70)

= (1/n2) ∗ {n ∗AvgV ar + n(n− 1) ∗AvgCov} (71)

=AvgV ar

n+(

1− 1n

)∗AvgCov. (72)

We can see that V ar(RP ) → AvgCov as n → ∞. The limiting portfolio standard deviationStDev(RP ) =

√AvgCov.

The historical volatility (standard deviation) of the return of a large stock is about 40% andthe typical correlation between the returns of large stocks is about 28%. On average, then,

AvgV ar = (0.40)2 = 0.16, (73)AvgCov = (0.40)(0.40)(0.28) = 0.0448, (74)

StDev(Rp) =√

0.16/n+ 0.0448(1− 1/n) −→ 21.17%. (75)

Note that when n = 20 the StDev(Rp) = 22.49%, which is very close to the limiting volatility.

What about a portfolio with arbitrary weights? We have:

σ2P = V ar(RP ) = Cov(RP , RP ) (76)

= Cov(∑i

xiRi, RP ) (77)

=∑i

xiCov(Ri, RP ) (78)

=∑i

xiσiσPCorr(Ri, RP ). (79)

Dividing both sides of this equation by σP yields this very important decomposition of thevolatility of a portfolio:

StDev(RP ) =∑i

Security i’s contribution to volatility of P︷ ︸︸ ︷xi︸︷︷︸

Amount of i held

∗ σi︸︷︷︸Total risk of i

∗ Corr(Ri, RP )︸ ︷︷ ︸Fraction of i’s risk common to P

(80)

Remark 4 Assuming that not all stocks have a perfect positive correlation of +1, we canconclude that when the weights are all positive, the volatility of the portfolio will be lowerthan the weighted average volatility given by

∑i xiσi. Note how the expected return of a

portfolio∑i xiE[Ri] is the weighted average of the individual returns. So, you can eliminate

some of the volatility by diversifying.

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3.6 Minimum variance portfolio of risky assets

The core optimization problem is formulated as follows:

P : Min V ar[R(x)] =12xTΣx (81)

subject to:x1 + x2 + · · ·+ xn = 1 (82)

In vector form, equation (82) may be expressed as

eTx = 1 (83)

where eT := (1, 1, . . . , 1) is the n-vector whose components are all ones.

Remark 5 The objective function is a (convex) quadratic function of the vector x and the oneconstraint is linear in x. So Problem P is called a quadratic programming problem, a specialcase of convex optimization problems. Quadratic programming problems are easy to solve usingcommercial software.

DefineL(x, λ) :=

12xTΣx− λ(eTx− 1). (84)

Here, λ is a real number. The function L(·) is called the Lagrangian for this problem.

Example 8 When n = 2 the core optimization problem is given by:

P : Min12{σ2

1x21 + σ2

2x22 + 2σ12x1x2} (85)

subject to:x1 + x2 = 1. (86)

The associated Lagrangian is given by:

L(x1, x2, λ) :=12{σ2

1x21 + σ2

2x22 + 2σ12x1x2} − λ(x1 + x2 − 1). (87)

Now consider the following unconstrained problem:

L(λ) : V (λ) := minx

L(x, λ). (88)

Let x(λ) denote an optimal solution to problem L(λ). The following is a fundamental theoremof convex analysis:

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Theorem 1

(1) If x∗ is an optimal solution to P, then there exists a λ such that x∗ = x(λ).

(2) If x(λ) satisfies the portfolio constraint (82), then x(λ) is an optimal solution to P.

Remark 6 The variable λ is called a dual variable. As in linear programming, there is onedual variable for each constraint. Note that there is only one dual variable here regardless ofthe number of x variables. It turns out that primal problem P has a dual problem given by

D : maxλV(λ), (89)

such that the optimal values of the primal and dual problems are equal. The dual problemis a concave optimization problem. Essentially, this duality extends the well-known duality oflinear programming.

We can use Theorem 1 to obtain a closed-form solution to our original problem P. Here isthe “game plan.”

Step 1. Let x∗ be an optimal solution to P and pick a λ such that x∗ = x(λ). (Theorem 1(a)guarantees that this is possible.) Since x(λ) is an optimal solution to (88), an unconstrainedproblem, we know from calculus that all n partial derivatives of L(·, λ) evaluated at x(λ) mustequal zero. These n equations in n unknowns may be solved to determine x(λ).

Step 2. It remains to pin down the value for λ, which requires one additional equation.Theorem 1(b) provides it, namely, the portfolio constraint (83).

We illustrate this game plan in the special case that n = 2. When presented in matrix form,the equations we derive all hold true for the general case.

First, the partial derivatives must vanish, i.e.,

∂L

∂x1= σ2

1x1 + σ12x2 − λ = 0. (90)

∂L

∂x2= σ2

2x2 + σ12x1 − λ = 0. (91)

(To simplify notation we suppress the dependence of x on λ.) In matrix form equations (90)-(91)may be represented as (

σ21 σ12

σ21 σ22

)(x1

x2

)= λ

(11

)(92)

or equivalently asΣx = λe. (93)

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These equations — keep in mind that there are n of them! — are called the first-order optimalityconditions. Now by multiplying both sides of (93) by Σ−1 we solve for the optimal x = x(λ) as:

x(λ) =

(x1(λ)x2(λ)

)= λ

(Σ−1e

). (94)

The vector x(λ) is proportional to the vector Σ−1e, with λ being the proportionality constant.Since we know there is some λ for which x∗ = x(λ), the optimal solution to P is also proportionalto the vector Σ−1e. Since the

∑i x∗i = 1, the proportionality constant is simply the reciprocal of

the sum of the components of Σ−1e, i.e., to arrive at the optimal solution x∗ from Σ−1e all onehas to do is simply “normalize” the latter vector by its sum. For example, suppose the vectorΣ−1e = (10, 20, 30). The sum is 60 and so the unique vector that is proportional to (10, 20, 30)and which sums to one is the vector (1/6, 1/3, 1/2). Finally, we observe that the multiplicationof Σ−1 by the vector e simply computes the vector of row sums of the matrix Σ−1.

Example 9 Let’s return to the data of Examples 5 and 7. Recall that we found that theminimum variance portfolio was (1/7, 6/7). In this example,

Σ =

(0.2500 0.02500.0250 0.0625

), (95)

and

Σ−1 =1

(0.25)(0.0625)− (0.025)2

(0.0625 −0.0250−0.0250 0.2500

)(96)

=1

0.015

(0.0625 −0.0250−0.0250 0.2500

)(97)

=

(4.16̄ −1.6̄−1.6̄ 16.6̄

). (98)

The vector of row sums associated with Σ−1 is (2.5, 15). The sum of these two numbers 17.5.Thus, the optimal portfolio vector is (2.5/17.5, 15/17.5) = (1/7, 6/7)!

Remark 7 It is not difficult to show that

λ =1

eTΣ−1e. (99)

In words, the value of λ can be obtained by merely summing up the values for all elements ofthe inverse of the covariance matrix, and taking the reciprocal. With a little more work, youcan show that λ actually equals the minimum variance! Returning to the example above, thesum of the elements of Σ−1 equals 17.5. The minimum variance equals 1/17.5 and the minimumstandard deviation equals

√1/17.5 = 23.9%, as before!

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Remark 8 If you are just interested in the minimum variance portfolio, you do not need tocalculate the determinant here as it is just a proportionality constant. (You have to normalizethe vector Σ−1e anyway, so the constant 1/0.015 is unnecessary.)

Example 10 Consider three risky assets whose covariance matrix Σ is

Σ =

216 70 −32470 25 −150

−324 −150 1596

. (100)

(Here, we measured returns in percents.) The inverse of the covariance matrix is

Σ−1 =

0.1480 −0.5367 −0.0204−0.5367 2.0388 0.0827−0.0204 0.0827 0.0043

. (101)

Thus,

Σ−1e =

−0.40811.59710.0661

(102)

eTΣ−1e = −0.4081 + 1.5971 + 0.0661 = 1.2379. (103)

The sum of all entries of Σ−1 is 1.2379 and this value equals the reciprocal of the minimum vari-ance. Dividing the vector (−0.4081, 1.5971, 0.0661) by 1.2379 yields (−0.3297, 1.2756, 0.0534),and this is the minimum variance portfolio. Its standard deviation is

√1/1.2379 = 80.78%.

4 Computing the Tangent Portfolio

In this section we assume the market includes a risk-free security that returns rf in all states.The tangent (or market) portfolio solves the following optimization problem:

(TP ) : Max

{R̄Tx− rf√xTΣx

: eTx = 1

}, (104)

where rf denotes the risk-free rate. The objective function is called the Sharpe ratio.

It can be shown that all mean-variance efficient portfolios must be a combination of therisk-free security and the tangent portfolio. That is, each investor should allocate a portion oftheir budget to the risk-free security and the remaining portion to the tangent portfolio. Thesub-allocations to the individual risky securities are determined by the weights x∗ that definethe tangent portfolio (to be determined). A very risk-averse investor allocates very little to the

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tangent portfolio, whereas an investor with a large appetite for risk may hold a negative portionin the risk-free security (i.e. borrow at the risk-free rate) to invest more than the budget intothe tangent portfolio.

We now turn to explaining how to calculate the tangent portfolio. Define

R̂ := R̄− rfe, (105)

which is the vector of excess returns obtained by simply subtracting the risk-free rate from eachof the asset’s expected returns. Problem (TP ) may be expressed as:

(TP ) : Max

{R̂Tx√xTΣx

: eTx = 1

}, (106)

Note that if one doubles the vector x the objective function in (TP ) remains the same. (Theobjective function is said to be “homogeneous of degree 0”.) Thus, we can ignore the constraintthat eTx = 1, because at the end we can perform the necessary normalization. (This assumesthat the sum of the xi is positive.)

Now imagine we knew the expected return R̄0 of the tangent portfolio. Let R̂0 := R̄0 − rfdenote the excess expected return of the tangent portfolio. The tangent portfolio would thenhave to satisfy the following mean-variance optimization problem:

Min{

12xTΣx : R̂Tx = R̂0

}. (107)

Inspect this optimization problem closely. It is the same as our original optimization problemP except that the constraint now has a vector R̂ that replaces the vector e and a value R̂0 thatreplaces the 1. From a structural standpoint, however, nothing has fundamentally changed,other than the “names to protect the innocent.”

Let µ denote the dual variable for this problem. The tangent portfolio necessarily satisfiesthe first-order optimality conditions

Σx∗ = µR̂. (108)

Consequently, the tangent portfolio must be proportional to Σ−1R̂, and thus it may be foundin the analogous way we found the minimum variance portfolio.

Example 11 Consider again our three asset data of the previous example. Suppose the ex-pected returns are (R̄1, R̄2, R̄3) = (18%, 10%, 8%), and suppose the risk-free rate is 3%. Theexcess expected returns are (18% − 3%, 10% − 3%, 8% − 3%) = (15%, 7%, 5%). The tangentportfolio is proportional to

Σ−1

1575

=

−1.63896.63460.2944

. (109)

15

Thus the tangent portfolio vector is (−0.3098, 1.2542, 0.0557). (Divide the right-hand side of(109) by 5.2901 = −1.6389 + 6.6346 + 0.2944.) It’s expected return is 7.4112 and its varianceequals 0.8338.

Remark 9 You can verify that the variance of the tangent portfolio equals µR̂0 and µ =(eTΣ−1R̂)−1. In the example, R̂0 = 7.4112 − 3 = 4.112 and µ = 1/5.2901 and so the varianceequals 4.112/5.2901 = 0.8338.

5 Security Market Line (SML)

The first-order optimality conditions (108) imply a relationship between a security’s expectedreturn and a measure of its risk. This relationship is called the Security Market Line (SML) andis the most widely used approach to determine an appropriate expected return for a security(or asset). The SML can also be used to decide whether an asset should be added to an existingportfolio and, if so, how to determine its weight in the new portfolio.

5.1 Derivation of SML from the first-order optimality conditions

LetxM = (xM1, xM2, . . . , xMn)

denote the tangent portfolio (of risky assets), and let

R̄M =∑i

xiM R̄i = R̄TxM

denote the expected return on the tangent portfolio. We will derive the SML from the first-orderoptimality conditions (108).

We begin by noting that the ith equation of (108) in non-matrix notation is simply∑j

σijxMj = µ(R̄i − rf ). (110)

Since σij = Cov(Ri, Rj), it follows from the properties of covariance that

µ(R̄i − rf ) =∑j

Cov(Ri, Rj)xMj = Cov(Ri,∑j

RjxMj) = Cov(Ri, RM ). (111)

Consequently,

R̄i = rf +1µCov(Ri, RM ). (112)

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It remains to pin down the value of µ. Substitute xM into (108) and multiply both sides byxM to obtain that

xTMΣxM = V ar(RM ) = µ(xTM R̂). (113)

Now we use the fact that

xTM R̂ = xTM (R̄− rfe) = xTM R̄− rf (xTMe) = R̄M − rf (114)

to derive that

V ar(RM ) = µ(R̄M − rf ) =⇒ 1µ

=R̄M − rfV ar(RM )

. (115)

From (112) and (115), we conclude that

R̄i = rf +Cov(Ri, RM )V ar(RM )

(R̄M − rf ). (116)

Define the parameter

βi :=Cov(Ri, RM )V ar(rM )

=ρiMσiσMσ2M

=ρijσiσM

. (117)

It is called the security’s beta. Using the parameter βi, we can rewrite (116) as

R̄i = rf + βi(R̄M − rf ) Security Market Line (SML) (118)

The term R̄M − rf is called the Market Premium. So, the Security Market Line states that

Security’s expected return = the risk-free rate + (security’s beta)*(Market Premium), (119)

or, equivalently,

R̂i︷ ︸︸ ︷Security’s excess expected return =

βi︷ ︸︸ ︷(security’s beta) ∗

R̂M︷ ︸︸ ︷(Market’s excess expected return).(120)

In the portfolio context it is the security’s β that is considered an appropriatemeasure of a security’s “risk.”

Remark 10 Observe that there is a linear relationship between a security’s expected returnand its risk (as measured by its β). This relationship is what economists call an equilibriumrelationship in the following sense: If a security’s expected return was higher (lower) thanwhat is predicted by the SML, then there would be more (less) demand for it, thereby raising(lowering) its price and ultimately lowering (raising) its expected return to place it back on theSML.

Example 12 Based on the past 50 years, the market premium is approximately 5%. With arisk-free rate of 2% the SML becomes R̄ = 0.02 + 0.05 ∗ β.

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• If β = 0, then R̄ = 0.02. So when a security is uncorrelated with the market, it only“deserves” an expected return equal to the risk-free rate. This is because you can diversifyaway any of its firm-specific risk with a large portfolio.

• If β = 1, then R̄ = R̄M . So when a security’s β equals one, then its expected returnshould equal the market portfolio’s expected return.

• Suppose β < 0, i.e., a security is negatively correlated with the market portfolio? The SMLtells us its expected rate of return should be lower than the risk-free rate! The reason forthis is that this security provides an additional benefit by lowering the standard deviationof the market portfolio due to its negative correlation. This security acts like a “hedge.”Investors will accept a lower expected rate of return for such a security because it providesa benefit for lowering overall portfolio risk.

Example 13 Stock A has a beta of 0.50 and Stock B has a beta of 1.25. Suppose rf = 4% andR̄M = 10%. What is the expected return of an equally weighted portfolio of these two stocks?Applying the SML, we have:

R̄A = rf + βAR̄M − rf = 4% + 0.50(10%− 4%) = 7%, (121)R̄B = rf + βBR̄M − rf = 4% + 1.25(10%− 4%) = 11.5% (122)

Expected return of an equally weighted portfolio is 0.50(7%) + 0.50(11.5%) = 9.25%.

What is the beta of a portfolio P?

βP =Cov(RP , RM )

V ar(RM )=Cov(

∑ixiRi, RM )

V ar(RM )=

∑ixiCov(Ri, RM )

V ar(RM )=∑

i

xiCov(Ri, RM )

V ar(RM )=∑

i

xiβi. (123)

We can see that the beta of a portfolio is the weighted average of the beta of the assets in theportfolio.

Example 14 Consider the data of the previous example. The portfolio’s beta is0.50(0.50) + 0.50(1.25) = 0.875. Applying the SML again, we have that

R̄P = rf + βP (R̄M − rf ) = 4% + (0.875)(10%− 4%) = 9.25%.

5.2 Estimation of β via regression

One can estimate β via linear regression. Its value corresponds to the slope of the best-fittingline in the plot of the security’s excess return versus the market’s excess return. Linear regressionmodels the excess return of a security as the sum of three components:

R̂i = αi + βiR̂M + εi. (124)

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• αi is the constant or intercept term of the regression, also called the stock’s alpha.

• βiR̂M represents the sensitivity of the stock to market risk.

• εi is the error (or residual) term. The average error is assumed to be zero.

Taking the expectations of both sides of (124), we have

E[R̂i] = βiE[R̂M ] + αi. (125)

The stock’s alpha, αi, measures the historical performance of the security relative to the ex-pected return predicted by the SML. It is a risk-adjusted measure of a stock’s historical perfor-mance. According to the SML, αi should not be significantly different from zero.

Example 15 Suppose you estimate that stock A has a volatility of 26% and a beta of 1.45,whereas stock B has a volatility of 37% and a beta of 0.79. Which stock has more total risk?Which stock has more market risk? Total risk is measured by volatility; therefore stock B hashigher total risk. Market risk is measured by beta; therefore stock A has higher market risk.

Suppose the risk-free rate is 3% and you estimate the market’s expected return as 8%.Which firm has a higher cost of equity capital? According to the SML:

E[RA] = rf + βA(E[RM ]− rf ) = 3% + 1.45(8%− 3%) = 10.25%. (126)E[RB] = rf + βB(E[RM ]− rf ) = 3% + 0.79(8%− 3%) = 6.95%. (127)

Market risk cannot be diversified. It is the market risk that determines the cost of capital. Weconclude that firm A has the higher cost of equity capital even though it is less volatile.

5.3 Derivation of SML from incremental analysis

Consider this question: You are given a portfolio P and an asset i not in portfolio P . Willadding asset i improve your portfolio? According to the theory, adding asset i (either witha positive or negative weight) to P will be beneficial if by doing so it will increase the (new)Sharpe ratio.

Example 16 Suppose portfolio P ’s expected return is 10%, its volatility is 20% and the risk-free rate is 2%. Suppose further that a particular mix of asset i and P yields a portfolio P ′

with an expected return of 9.5% and a volatility of 15%. Is asset i beneficial? The Sharpe ratioof P is (0.10-0.02)/0.20 = 0.40 and the Sharpe ratio for P ′ is (0.095-0.02)/0.15 = 0.50.

Here is what we can do. For every dollar invested in P ′ we can borrow $1/3 at the risk-freerate and invest the proceeds in P ′. The revised portfolio P ′′ has a weight of 4/3 on P and a

19

weight of −1/3 on the risk-free asset. The volatility of P ′′ is 4/3*the volatility of P ′ = 4/3(0.15)= 20% and its expected return is (4/3)(9.5) + (-1/3)2 = 12%. By adding asset i and borrowing,we have constructed a new portfolio P ′′ whose volatility is the same as the original portfolio P ,yet has an expected return that is 2% higher. Clearly, asset i is beneficial.

Consider modifying portfolio P , as follows: for every dollar invested in P we will either (a)borrow x > 0 dollars at the risk-free rate and invest the proceeds in asset i, or (b) short −x > 0dollars of asset i and invest the proceeds at the risk-free rate. The revised portfolio P x will stillhave a weight of 1 on P , but will now have a weight of x on asset i and a weight of −x on therisk-free asset. The expected return and variance of P x are

E[RPx ] = R̄P + x(R̄i − rf ) = R̄P + xR̂i (128)V ar[RPx ] = σ2

P (1)2 + σ2i x

2 + 2(1)(x)σiP = σ2P + 2σiPx+ σ2

i x2, (129)

and so the Sharpe ratio SR(x) associated with P x is

SR(x) =R̂P + R̂ix√

σ2P + 2σiPx+ σ2

i x2

:=R(x)√V ar(x)

. (130)

With this notation, note that SR(0) equals the Sharpe ratio for P ,R(0) equals R̂P and√V ar(0)

equals the volatility of P .

Since the value of x can be positive or negative, the Sharpe ratio for P x will be higher thanthat for P if the derivative of SR(x) at x = 0 is not zero. For example, if SR′(0) > 0, thenSR(x) > SR(0) for sufficiently small values of x > 0; or if SR′(0) < 0, then SR(x) > SR(0)for sufficiently small values of x < 0. So, if the Sharpe ratio cannot be increased with asset i,then it must necessarily be the case that SR′(0) = 0. Conversely, the condition SR′(0) = 0 isalso sufficient, namely, if it holds, then the Sharpe ratio cannot be increased with asset i.

Now for the calculus. Recall that

d

dx(f(x)/g(x)) =

f ′(x)g(x)− f(x)g′(x)g(x)2

, (131)

and so

SR′(x) =R′(x)

√V ar(x)−R(x) V ar′(x)

2√V ar(x)

V ar(x)(132)

SR′(0) =R̂iσP − R̂P 2σiP

2σP

σ2P

. (133)

Since σ2P > 0,

SR′(0) 6= 0⇐⇒ R̂iσP 6=R̂PσiPσP

⇐⇒ R̂i 6=σiPσ2P

R̂P = βPi R̂P , (134)

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from which we may conclude that a necessary and sufficient condition for not being able toimprove the Sharpe ratio of P with asset i is that

R̄i = rf + βPi (R̄P − rf ) := required return for asset i. (135)

We can think of the right-hand side of (135) as the required return for asset i to compensate forthe risk it will contribute to portfolio P . If R̄i is higher than its required return, then addingasset i to P with some positive weight will increase the Sharpe ratio; if R̄i is lower than itsrequired return, then adding asset i to P with some negative weight will increase the Sharperatio.

Remark 11 The derivation above will apply if asset i already belongs to the portfolio P withsome non-zero weight. If asset i’s expected return does not equal its required return, this meansthat the existing weight xi in portfolio P can be adjusted (up or down) to improve the Sharperatio. When portfolio P is mean-variance efficient, its Sharpe ratio cannot be improved, whichimplies that existing portfolio weights cannot be adjusted without lowering the Sharpe ratio.

Example 17 You are currently invested in a broad-based fund with an expected return of 15%and a volatility of 20%. Your broker suggests that you add a real estate fund to your portfolio.It has an expected return of 9%, a volatility of 35% and a correlation of 0.10 with your existingfund. Will adding the real estate fund improve your portfolio? The risk-free rate is 3%.

Let P denote the portfolio consisting of the existing fund and let i denote the real estatefund. Here,

βPi = ρiPσi/σP = 0.10(0.35)/(0.20) = 0.175, (136)

and sorf + βPi (R̄P − rf ) = 3% + 0.175(15%− 3%) = 5.1%. (137)

As 5.1% < 9%, it will certainly be advantageous to invest some amount in the real estate fundto improve your Sharpe ratio of (0.15-0.03)/0.20 = 0.60.

Example 18 Consider the data for the previous example. What is the optimal choice of x?Here, σiP = (.35)(0.20)(0.1) = 0.007, and so the Sharpe ratio (130) is

SR(x) =0.12 + 0.06x√

0.04 + 0.014x+ 0.1225x2. (138)

You can try different examples for x in increments of 0.01 and choose the best. For example,SR(0.10) = 0.6103. It turns out that x = 0.11 is the best and yields a Sharpe ratio of 0.6104.So, if you had $10,000 in your fund, you can borrow $1,100 at the risk-free rate and invest inthe real estate fund. If you did not want to borrow, you would have to sell a portion of yourexisting broad-based fund. In this case, the portfolio weights are 1/(1 + x) and x/(1 + x) onthe broad-based and real-estate funds, respectively. The amount invested in the broad-basedfund will equal $10,000/(1 + x) = $9,009, so you will sell $991 of the existing fund and buythis amount of the real estate fund.

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Remark 12 It is possible to determine the value of x in our example without having to enu-merate all possibilities. As you add asset i to the portfolio P , the βP

x

i will change, since asseti now belongs to the new portfolio. We should stop adding asset i precisely when SR′(x) = 0,which will happen exactly when R̂i = βP

x

i R̂Px . It is important to keep in mind here that bothβP

x

i and R̂Px depend on the value of x.

Recall that the return of portfolio P x is RP + x(Ri − rf ), and so

R̂Px = R̂P + R̂ix; (139)

moreover,

βPx

i =Cov(Ri, RP + x(Ri − rf ))V ar(RP + x(Ri − rf ))

=σiP + σ2

i x

σ2P + 2σiPx+ σ2

i x2. (140)

Consequently,

SR′(x) = 0⇐⇒ R̂i =σiP + σ2

i x

σ2P + 2σiPx+ σ2

i x2(R̂P + R̂ix). (141)

An inspection of (141) shows that the quadratic term in x will cancel, thereby leaving just alinear expression that can easily be solved. After a little algebra, we have

x∗ =σ2P R̂i − σiP R̂Pσ2i R̂P − σiP R̂i

. (142)

Returning to the previous example, σ2P = (0.20)2 = 0.04, σ2

i = (0.35)2 = 0.1225, σiP =(0.20)(0.35)(0.10) = 0.007, R̂P = 0.15− 0.03 = 0.12 and R̂i = 0.09− 0.03 = 0.06, and so

x∗ =(0.04)(0.06)− (0.007)(0.12)

(0.1225)(0.12)− (0.007)(0.06)= 0.109244 = 11%. (143)

6 CAPM Assumptions and Implications

There are three assumptions underlying the CAPM:

• Investors can buy or sell all securities at competitive market prices (without transactioncosts) and can borrow and lend at the risk-free rate.

• Investors hold only efficient portfolios of traded assets. Such portfolios yield the maximumexpected return for a given level of volatility.

• Investors have the same (homogeneous) expectations regarding the expected returns,volatilities and correlations of the securities.

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These assumptions imply the following conclusions. First, each investor will identify thesame portfolio of risky assets that has the highest Sharpe ratio. This is the tangent portfolioof risky assets. The only difference between each investor is the amount of money investedin the risk-free asset, which depends on their respective appetites for risk. For example, oneinvestor may choose to invest 50% of his wealth in the risk-free rate and the remaining 50% inthe tangent portfolio. The actual weight of this investor’s portfolio in risky asset i is simply0.50x∗i . Another investor may choose to invest only 10% of her wealth in the risk-free asset;the actual weight of this investor’s portfolio in risky asset i is 0.90x∗i . The sum of all investor’sportfolios must equal the portfolio of all risky securities in the market, which is commonlyreferred to as the market portfolio. (It is often proxied by the S&P 500.) Therefore, the tangentportfolio of risky securities equals the market portfolio. This statement can be viewed as sayingthat “supply equals demand.” As we alluded to before, prices in the market adjust so that allsecurities are held in the tangent (market) portfolio in just the right amounts determined bythe SML.

In sum, according to the assumptions of CAPM, all mean-variance efficient portfolios mustlie on the Capital Market Line (CML) defined by:

R̄P = rf +

(R̄M − rfσM

)σP . (144)

That is, a portfolio P is efficient if and only if it has the same Sharpe ratio as the market’s,which is just another way of saying that all mean-variance efficient portfolios must be somecombination of the risk-free asset and the market (tangent) portfolio.

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7 Summary of key definitions, notation and formulae

Risk-free rate = rf

Random return on asset i = Ri

Expected return on asset i = E[Ri] = R̄i

Expected excess return on asset i = R̄i − rf = R̂i

Standard deviation of return on asset i = σi

Covariance between returns on assets i and j = σij

Covariance matrix = Σ

Correlation between returns on assets i and j = ρij = σij/σiσj

Portfolio weight on asset i = xi

Portfolio random return = RP = R(x) =∑

i

xiRi

Portfolio expected return =∑

i

xiR̄i

Portfolio expected excess return =∑

i

xiR̂i

Portfolio variance V ar(RP ) = xT Σx =∑

i

∑j

σijxixj

Portfolio volatility StDev(RP ) =√V ar(RP )

Portfolio variance for two assets = σ2i x

2i + σ2

jσ2j + 2xixjσiσjρij

Minimum variance portfolio of two assets = (x∗1, x∗2) =

(σ22 − σ12

σ21 + σ2

2 − 2σ12,

σ21 − σ12

σ21 + σ2

2 − 2σ12

)Variance of a large portfolio =

AvgV ar

n+

(1−

1

n

)∗AvgCov

StDev(RP ) =∑

i

Security i’s contribution to volatility of P︷ ︸︸ ︷xi︸︷︷︸

Amount of i held

∗ σi︸︷︷︸Total risk of i

∗ Corr(Ri, RP )︸ ︷︷ ︸Fraction of i’s risk common to P

Minimum variance portfolio = x(λ) =

(x1(λ)x2(λ)

)= λ(

Σ−1e),

Minimum variance = λ =1

eT Σ−1e=

1

sum of the entries of Σ−1

Sharpe ratio for a portfolio =R̄T x− rf√xT Σx

=Portfolio’s expected excess return

Portfolio’s volatility

Tangent portfolio = x(µ) =

(x1(µ)x2(µ)

)= µ(

Σ−1R̂), µ =

1

eT Σ−1R̂

Security beta βi =Cov(Ri, RM )

V ar(rM )=ρiMσiσM

σ2M

=ρijσi

σM

Security market line (SML): R̄i = rf + βi(R̄M − rf ) = Required return on asset i

Capital market line (CML): R̄P = rf +

(R̄M − rfσM

)σP

24